Square Root of 100000

The square root is the inverse of the square of a number. 100000 is not a perfect square. The square root of 100000 is expressed in both radical and exponential form. In the radical form, it is expressed as √100000, whereas (100000)^(1/2) in the exponential form. √100000 = 316.227766, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the Prime factorization of a number. Now let us look at how 100000 is broken down into its prime factors:

Step 1: Finding the prime factors of 100000 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5: 2^5 x 5^4

Step 2: Now that we have found the prime factors of 100000, the second step is to make pairs of those prime factors.

Since 100000 is not a perfect square, calculating its square root using prime factorization directly is not feasible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 100000, we need to group it as 00, 00, and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is ‘3’ because 3 × 3 = 9, which is less than 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down the next pair of digits (00) to the right of the remainder, making it 100.

Step 4: The new divisor will be the sum of the old divisor and the quotient, which is 6 (3 + 3). Now we need to find n such that 6n × n ≤ 100.

Step 5: Let n be 1, then 61 × 1 = 61, and subtract 61 from 100 to get 39.

Step 6: Bring down the next pair of digits (00) to make it 3900.

Step 7: The new divisor is 62 (61 + 1), and we need to find n such that 62n × n ≤ 3900.

Step 8: Let n be 6, then 626 × 6 = 3756, and subtract 3756 from 3900 to get 144.

Step 9: Add a decimal point and bring down two zeros, making it 14400.

Step 10: Continue this process until you reach the desired decimal places.

So the square root of √100000 is approximately 316.2277.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 100000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √100000. The smallest perfect square less than 100000 is 96100 (310^2), and the largest perfect square greater than 100000 is 102400 (320^2). √100000 falls somewhere between 310 and 320.

Step 2: Now we need to estimate the value between these two perfect squares. We can see that 100000 is closer to 102400 than it is to 96100, so a value closer to 316 is reasonable.

Step 3: Using the approximation, we find the square root of 100000 to be approximately 316.2277.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction; its decimal goes on forever without repeating. √100000 is irrational.
   
• Principal square root: This refers to the positive square root of a number. For example, the principal square root of 100000 is approximately 316.23.
   
• Prime factorization: A way of expressing a number as the product of its prime factors. For instance, the prime factorization of 100000 is 2^5 × 5^4.
   
• Perimeter: The total length around a two-dimensional shape. For example, the perimeter of a rectangle is calculated as 2 × (length + width).